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Wave slamming loads on a circular cylinder during water entry and the subsequence submergence process are predicted based on a numerical wave load model. The wave impact problems are analyzed by solving Reynolds-Averaged Navier-Stokes (RANS) equations and VOF equations. A finite volume approach (FV) is employed to implement the discretization of the RANS equations. A two-dimensional numerical wave tank is established to simulate regular ocean waves. The wave slamming problems are investigated by deploying a circular cylinder into waves with a constant vertical velocity. The present numerical method is validated using other numerical or theoretical results in accordance with varying free surface profiles when a circular cylinder sinks in calm water. A numerical example is given to show the submergence process of the circular cylinder in waves, and both free surface profiles and the pressure distributions on the cylinder of different time instants are obtained. Time histories of hydrodynamic load on the cylinder during the submergence process for different wave impact angles, wave heights, and wave periods are obtained, and results are analyzed in detail.

Wave impact problem is of great interest in the marine and offshore industries, especially for subsea working systems which are widely used in such applications as marine resource development and utilization, maritime exploration and survey; the important problem which needs to be solved before their proper operations is to accurately study the phenomenon of offshore structures impacting ocean wave and the following submergence process. The process of subsea structures lowering through wave zone accompanies the interaction between the air, wave, and the solid body, which is a complicated fluid-solid problem, involving consideration of time-varying hydrodynamic forces (slamming, drag, inertia, and buoyancy) and time-varying waves. In hostile ocean conditions, these forces can result in significant localized and even catastrophic structural damage on structures ranging from deployed structures to deploying equipment and heave compensator systems and so forth. Therefore, the accurate prediction of wave slamming loads and time histories of hydrodynamic force, as well as the sensitivity of these loads to wave parameters, is of significant importance.

Researches on water entry and wave impact problems were firstly carried out by Karman [

For a rigid cylinder, pioneering studies applied several different methods, including flat plate theories, generalized Wagner theory [

This work is devoted to investigate the complex interaction between a circular cylinder and waves during the submergence process. Firstly, both governing equations and boundary conditions are outlined; to study the interaction between wave and cylinders, a numerical wave tank is established, and numerical wave generation and absorption method are presented. Secondly, snapshots of the simulation of calm water entry of a circular cylinder are compared with those of experiments by previous researches. Then, numerical simulations of a circular cylinder impact with waves are conducted, and free surface profiles of different time instants of submergence process are shown. Finally, slamming loads on the circular cylinder are computed, and influence of wave parameters is discussed.

In this work, governing equations for the CFD calculations are RANS equations for homogeneous, incompressible fluid flows, and they are written as follows:

The empirical coefficients in (

The complex free surface is tracked by the VOF method, and to accomplish the capturing of the interface between the air phase and the water phase, a continuity equation for the fraction of volume of water is solved, which has the following form:

The sum of the volume fraction of water and air is given by

To solve governing equations, it is necessary to specify appropriate boundary conditions at all boundaries of the domain. The boundary conditions which need to be satisfied are as follows: (

Numerical wave tank and a moving circular cylinder.

In order to investigate wave loads on a circular cylinder during its crossing air-water interface, it is necessary to establish a numerical wave tank. In this paper, a piston type wave-maker is utilized to simulate ocean waves, and a sketch of a numerical wave tank with a piston wave-maker located at the left boundary of the domain is illustrated in Figure

The plunger moves horizontally with a sinusoidal function:

Equation (

And wave height

As mentioned in Section

The 2D numerical tank is a rectangle with 40 m height, and the length of which is determined by the target waves. The calm water depth (

Schematic diagrams of mesh.

General diagram of mesh

Mesh surrounding moving body

Mesh near the wave-maker

A finer mesh can be observed near the free surface for the possibility of wave breaking and near the moving wave-maker, while coarser meshes can be found towards the right wall and the bottom of domain.

A commercial CFD software ANSYS Fluent was employed to solve the RANS equation with the free surface capturing VOF scheme and turbulence equations. Setups of models in the present simulations are briefly introduced as follows: georeconstruct scheme, which is the default scheme of the VOF model, is applied to compute the air-water interface (wave surface). For pressure-velocity coupling, the pressure implicit with splitting of operators (PISO) scheme is used, because problems presented in this paper generally involve transient flows. PRESTO! scheme is used for pressure interpolation scheme of VOF two-phase models. The second-order upwind scheme is applied for discretization of the momentum equation to obtain second-order accuracy. As far as the time step size (

A numerical experiment for simulating a target wave (wave height = 2.5 m, wave period = 6 s) is conducted to verify the versatility and correctness of the numerical wave generation and damping methods presented in Section ^{2};

Four wave elevation probes are placed at locations of

Wave elevation at four different locations: (a)

To reflect the capability of the presented numerical model to deal with the problem of a body moving near the air-water interface, the example of a circular cylinder sinking in calm water is used. The setup of the numerical model is given below. The radius of the circular cylinder ^{2}, and the cylinder moves downwards with a constant velocity

Comparisons between free surface profiles of the present model and previous researches, which include Lin’s and Greenhow and Moyo’s numerical results and Tyvand and Miloh’s theoretical results [

Comparisons of results by different models of a circular cylinder sinking in calm water: the present results (red solid line); Lin’s results (blue solid line); Greenhow et al.’s results (dashed line); Tyvand et al.’s results (dotted line).

The choice of mesh can affect the accuracy of simulation results, computational efficiency, and the solution stability and is dependent on amount of memory available. Therefore, it is important to discuss the size and quality of mesh and choose appropriate meshes. Four different mesh sizes are employed to perform the mesh size study, and main differences between them are sizes of finer meshes near the circular cylinder, the wave-maker, and the free surface, while coarser meshes are basically the same. More information of meshes can be found in Table

Mesh sizes used for mesh size study.

Mesh size | Number of cells | Mesh size near the circular cylinder | Mesh size near the free surface |
---|---|---|---|

Mesh 1 | 26160 | 0.07854 m ( | 0.2 m ( |

Mesh 2 | 36388 | 0.05236 m ( | 0.133 m ( |

Mesh 3 | 51590 | 0.03927 m ( | 0.1 m ( |

Mesh 4 | 57460 | 0.03142 m ( | 0.08 m ( |

Main setups of simulations are as follows: the target wave which is simulated in Section

Figure

Hydrodynamic forces on the circular cylinder with different mesh size.

An example is utilized to show the whole water entry process of a circular cylinder in waves, main setups of this numerical example are the same as those described in Section

The free surface profiles and pressure distributions at

Submergence process of a circular cylinder in waves: (a)

Pressure distributions on the cylinder: (a)

At

During the transit of a circular cylinder through air-water interface, an important parameter is the vertical hydrodynamic force on the cylinder, also known as the slamming load. The slamming coefficient can be defined, which is the nondimensional impact force:

When

As shown in Figure

Vertical hydrodynamic force on a circular cylinder.

Figure

Wave impact phase angles can reflect the relative position between the circular cylinder and wave surface when impact occurs. Figure

Wave impact positions.

To investigate the influence of wave impact phase angles, positions marked as 1, 2, 3, and 4 shown in Figure

Two groups of theoretical impact time

30.684 s, 26.184 s, 27.684 s, and 29.184 s

36.684 s, 32.184 s, 33.684 s, and 35.184 s

Obviously, impact times of the second group are obtained by those of first group plus period of wave, 6 s in this case. Figures

Time history of hydrodynamic force with different impact phase angles: (a)

Velocity vectors when water impact occurs with different impact phase angles: (a)

From Figures

Firstly, the simulated impact time is a little earlier or later than the theoretical one, which is due to calculation and simulation error; it is worth noting that no slamming would occur for case of

Secondly, the time histories of hydrodynamic forces with the same impact phase angles are mostly identical, which illustrates that smaller impact time can be applied to save simulation time as long as the first wave periods are avoided.

Thirdly, after slamming, the hydrodynamic forces gradually increase to the maximum value for all four phase angles, which is equal to about 40 kN, and this phenomenon indicates the impact wave phase exerts no influence on the maximum hydrodynamic force; considering the length of the time interval between impact time

Impact time and maximum hydrodynamic force of different phase angles.

Impact phase angle | | | | |
---|---|---|---|---|

Time of impact ( | 36.64 s | 32.37 s | 33.68 s | 35.13 s |

Time of | 41.18 s | 36.33 s | 36.39 s | 36.8 s |

| 4.56 s | 3.96 s | 2.71 s | 1.67 s |

Fourthly, after it reaches the maximum value, the hydrodynamic force oscillates near an equilibrium point about 31 kN, which is equal to the buoyancy of the circular cylinder, and the oscillation period is about 6 s, which is equal to period of the target wave; besides, as simulation time goes on, the oscillation amplitude is smaller and smaller and hopefully becomes zero eventually, mainly because of the fact that wave effect becomes smaller with water depth increasing.

As far as the slamming force are concerned, cases with phase angle

In order to study the influence of the wave height, time histories of hydrodynamic force with five different wave heights of regular waves, 1.7 m, 2 m, 2.3 m, 2.6 m, and 3 m, are considered; other parameters are as follows: impact phase angle =

From Figure

Fluid forces with different wave heights.

In order to investigate the influence of the wave period, time histories of fluid forces with five different wave periods of regular waves, 5 s, 6 s, 7 s, 8 s, and 9 s, are considered, which are shown in Figure

Fluid forces with different wave periods.

The following phenomena can be seen from Figure

For the circular cylinder that is moving through wave surface, the Froude number

Figure

Fluid forces with different cylinder radiuses.

Fluid forces with different vertical velocities.

(

(

(

(

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the financial support of Natural Science Foundation of Hunan Province of China (Grant no. 2017JJ3393), the National Natural Science Foundation of China (no. 51305463), and the National Key Research and Development Plan of China (no. 2016YFC0304103).